[Paper Review] Polyhedral Surfaces in Wedge Products

Thilo Rörig, Günter M. Ziegler|ArXiv.org|Aug 21, 2009

Advanced Combinatorial Mathematics7 references4 citations

TL;DR

This paper introduces the wedge product of polytopes as a new construction that generalizes subdirect products and dualizes wreath products. It shows that wedge products of polygons and simplices contain combinatorially regular polyhedral surfaces of type {p,2q}, which can be projected into R³ with high-dimensional realization spaces, yielding at least 6p moduli for the case {p,4}, enabling rich geometric deformations and dual surface realizations via duality.

ABSTRACT

We introduce the wedge product of two polytopes. The wedge product is described in terms of inequality systems, in terms of vertex coordinates as well as purely combinatorially, from the corresponding data of its constituents. The wedge product construction can be described as an iterated ``subdirect product'' as introduced by McMullen (1976); it is dual to the ``wreath product'' construction of Joswig and Lutz (2005). One particular instance of the wedge product construction turns out to be especially interesting: The wedge products of polygons with simplices contain certain combinatorially regular polyhedral surfaces as subcomplexes. These generalize known classes of surfaces ``of unusually large genus'' that first appeared in works by Coxeter (1937), Ringel (1956), and McMullen, Schulz, and Wills (1983). Via ``projections of deformed wedge products'' we obtain realizations of some of the surfaces in the boundary complexes of 4-polytopes, and thus in R^3. As additional benefits our construction also yields polyhedral subdivisions for the interior and the exterior, as well as a great number of local deformations (``moduli'') for the surfaces in R^3. In order to prove that there are many moduli, we introduce the concept of ``affine support sets'' in simple polytopes. Finally, we explain how duality theory for 4-dimensional polytopes can be exploited in order to also realize combinatorially dual surfaces in R^3 via dual 4-polytopes.

Motivation & Objective

• To develop a new construction—wedge products of polytopes—that generalizes subdirect products and dualizes wreath products.
• To demonstrate that wedge products of a p-gon and a (q−1)-simplex contain combinatorially regular polyhedral surfaces Σp,2q of type {p,2q}.
• To provide geometric realizations of these surfaces in R³ via projections of deformed wedge products.
• To establish high-dimensional realization spaces (moduli) for such surfaces using affine support sets in simple polytopes.
• To extend the construction to realize dual surfaces Σp,4* of type {4,p} via duality in 4-polytopes.

Proposed method

• Define the wedge product of two polytopes via explicit linear inequality systems, enabling precise control over facet and vertex structure.
• Construct the wedge product C_p ⋈ Δ_{q−1} as an iterated subdirect product, with explicit vertex coordinates and combinatorial descriptions.
• Identify a specific subcomplex Σp,2q in the 2-skeleton of C_p ⋈ Δ_{q−1} that forms a regular, equivelar surface of type {p,2q}.
• Use Perles' lemma to project high-dimensional wedge products into R⁵, and for q=2, project further into R³ via orthogonal projection to the lower hull.
• Introduce the concept of affine support sets in simple polytopes to bound the dimension of realization spaces from below.
• Leverage duality in 4-polytopes to realize dual surfaces Σp,4* of type {4,p} in R³ through dual projections.

Experimental results

Research questions

• RQ1 Can the wedge product construction yield new families of regular polyhedral surfaces with high genus and symmetric structure in R³?
• RQ2 How many independent deformations (moduli) do the realizations of these surfaces admit, and can this be quantified?
• RQ3 Can the surfaces Σp,4 be realized in R³ via projection of a 4-polytope, and does the projection preserve the surface structure?
• RQ4 What is the role of affine support sets in estimating the dimension of realization spaces of projected polytopes?
• RQ5 Can duality in 4-polytopes be used to realize combinatorially dual surfaces Σp,4* in R³?

Key findings

• The wedge product of a p-gon and a 1-simplex (edge) yields a 2p-dimensional simple polytope W_{p,1} with a 2-skeleton containing a regular surface Σp,4 of type {p,4}.
• The surface Σp,4 can be projected orthogonally to R³ while preserving its structure, yielding a realization in 3D space.
• The realization space of Σp,4 has at least 6p independent moduli, established via an affine support set of size 2p in W_{p,1}.
• The dual surface Σp,4* of type {4,p} is realizable in R³ by projecting the dual 4-polytope, leveraging duality in 4-dimensional polytopes.
• For q>2 and p>3, no such projection of the surface Σp,2q from W_{p,q−1} to R⁴ exists due to topological obstructions, leaving the case p=3 open.
• The use of affine support sets provides a general method to lower-bound the number of moduli in projected realizations of polyhedral surfaces.